E ects of breaking vibrational energy equipartition on ...

Detection of gravitational wavesOn “Effective Temperatures”

Discussion and open questions

Effects of breaking vibrational energyequipartition on measurements of

temperature in macroscopic oscillators

L. Rondoni, Politecnico Torino

R. Belousov, M. Bonaldi, L. Conti, P. De Gregorio, C. Giberti

Firenze – 29 May 2014

PRL 2009; J. Stat. Mech. 2009 and 2013;Class. Quant. Grav. 2010; PRB 2011; PRE 2011 and 2012

http://www.rarenoise.lnl.infn.it/

L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators



Outline

1 Detection of gravitational wavesLangevin equationFluctuations

2 On “Effective Temperatures”1-dimensional models: comparison with experimentSimulations and “theory”Results

3 Discussion and open questions




Langevin equationFluctuations

Thermal fluctuations unobservable in macroscopic objects?

General Relativity predicts gravitational waves (GW): e.g.accelerating binary systems of neutron stars or black holes;vibrations of black holes or neutron stars.Hulse-Taylor measurement of orbits of two neutron stars, spirallingas if losing energy by GW emission; in excellent agreement withpredictions, were awarded Nobel prize in 1993.GW: kind of space-time ripples, in two fundamental states ofpolarization, cross and plus. Effect of GW on matter:squeezing and stretching, depending on phase.

plus

cross


Concise historyWeakest assumptions approach

News from RareNoise

The idea which was behind the RareNoise project

Ground-based Detectors

Can detect thermal fluctuations intrinsic to the test mass.

Expected to approach the quantum limit in the future.

Nonequilibrium stationary states and noise

Past studies had assumed the noise be Gaussian. However theexperimentalists’ interest is in the tails of the distributions.There, they may be not.

Then the question

We detect a rare burst. Is it of an external source? Or falsepositive due to rare nonequilibrium (and non-Gaussian)fluctuations? Knowing correct statistics is mandatory.

L. Rondoni, Politecnico Torino & P. De Gregorio, INFN Padova Deterministic Approach




Gravitational Wave detectorM otivation: GWs will provide new and unique information about astrophysical processes

GW amplitude:

A detection rate of few events/year requires

sensitivity of over timescales as short as 1msec

small signal noise noise sources must be reduced to very low levels

L. Conti - Bologna 17.03.2014L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators




GW detectors (interferometers)

LIGO @Livingston (USA)VIRGO @Cascina (Italy)

GEO600 @Hannover (Germany) TAM A300 @Tokyo (Japan)





GW detector noise budgetDominant Sources of Noise:• seismic noise• thermal noise• photon shot noise

ThermalShot

Seismic

sens

itivi

tyin

crea

ses





Thermal compensation

666 m687 m

Applied thermal gradient deforms the mirror and corrects the ROC

to correct mismatch of the mirrorRadius Of Curvature (ROC) due to:

fabricationthermal lensingthermo-elastic deformation

What is the ‘thermal noise’ of such a non-equilibrium body?L. Conti - Bologna 17.03.2014L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators




Resonant-bar GW detectors: feedback cooling down to mK:viscous force reduces thermal noise on length of resonant-bardetector AURIGA (PRL top ten stories, 2008).

Steady state modelled by 3 electro-mechanical oscillators withstochastic driving.





LdIs(t)

dt+ Is(t) [R + Rd ] +

qs(t)

C=√

2kBT0R Γ(t)

Id (t) = GIs(t − td )

td =π

2ωr

G � 1

Rd = GωrLin expresses viscous damping due to feedback;

No time reversal invariance (q′s = qs , I′s = −Is , t ′ = −t),

violates Einstein relation, but formally identical to equilibriumoscillator at fictitious temperature Teff = T0/(1 + g)

with feedback efficiency g = Rd/R, so that: 〈I 2s 〉 = 2k

BTeff/L

Hence, usually treated as equilibrium system!





PDF and fluctuation relation of injected power Pτ : Farago, ’02

ρ(ετ ) = limτ→∞

1

τln

PDF(ετ )

PDF(−ετ )=

{4γετ , ετ <

13 ;

γετ

(74 + 3

2ετ− 1

4ε2τ

), ετ ≥ 1

3 .

ετ = PτL/(kBT0R) =; γ = (R + Rd )/L, Teff = (22± 1) mK

secondderivativeof PDF;

b) ρ(ετ ).

Data fromMay 2005 toMay 2008.

Shades = experimental uncertainty on τeff, Teff, T0.


Concise historyWeakest assumptions approach

News from RareNoise

RN aluminum exp. - longitudinal and flexural oscillations

longitudinal flexural

L. Rondoni, Politecnico Torino & P. De Gregorio, INFN Padova Deterministic Approach



1-dimensional models: comparison with experimentSimulations and “theory”Results

For macroscopic systems in local thermodynamic equilibrium (LTE)

“the properties of a ‘long’ metal bar should not depend on whetherits ends are in contact with water or with wine ‘heat reservoirs’ attemperature T1 and T2” (Rieder, Lebowitz, Lieb, JMP 1967)

But modelling by 1-dimensional systems incurs in violations ofconditions of LTE, hence strong dependence on details ofmicroscopic dynamics: care must be taken in tuning parameters toobtain

“proper thermo-mechanical” behaviour.

Wanted “realistic” equilibrium properties:

thermal expansion, and temperature dependentelasticity, resonance frequencies and quality factor.

and non-equilibrium: linear “temperature” profile.





Deterministic reversible semi-openNose-Hoover at T1 and T1 + ∆T

Nearest- and next-nearest-neighbors L-J. N = 128, 256, 512

V (ri , ri±`) = ε

[(`r0

|ri − ri±`|

)12

− 2

(`r0

|ri − ri±`|

)6]

; ` = 1, 2

mri = F inti (ri , ri±1, ri±2)− χi ri

χi =m

τ2

(K

kBTi− 1

); Ki = mr2

i

for i = 1, 2 and N − 1,N

χi = 0 for i 6= 1, 2,N − 1,N

Looks more like 3D-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.5 1 1.5 2 2.5 3

V(x)[ǫ]

x = r/r0





Canonical and local canonical appear consistent with observedresults from simulations (elasticity etc.)

Kinetic temperature profile straightapart fromthermostattedborders,i = 1, 2,N − 1,N

Maybe better mixing?





Spectral density - Experiment and Simulations

For given z = z(t) real,

Sz (ω) =∫ +∞−∞ eiωt〈z(t)z(0)〉dt

e.g. z → x(t) = L(t)− 〈L〉,or z → v(t) = x(t); z → V (t)

10−4

10−3

10−2

10−1

100

101

102

0 5 10 15 20 25

ω1Sx(ω

)/r

20

ω/ω1

✲

10−3

10−1

101

0.6 1 1.4





Equilibrium & Nonequilibrium thermo-elasticity - Exp+Sim

T1 T1 + ∆T

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.004 0.008 0.012 0.016 0.02 0.024

(ωr

ω0

) 2 LJ

E [ ǫ ]

��

��

��

�

116

118

120

122

124

126

128

130

0.06 0.07 0.08 0.09 0.1 0.11

E[

ε

R0

]

kBT [ ε ]

�

�

�

�

4

4





Equilibrium & Nonequilibrium thermo-elasticity

1) 1D model reproduces thermo-elastic properties at equilibrium,e.g. linearity of elastic modulus E or of ωres with T ;

2) It works out of equilibrium as well:e.g. ωr = ωr (T ), with average temperature T = (T1 + T2)/2, andωr (T ) the equilibrium resonance frequency.

3) Non trivial: for larger ∆T , theory does not apply. Explanationin terms of local canonical,

ψi = exp (−Ei/kBTi )

i.e. under local equilibrium.





Experiment: low-loss (high quality factor) bar.⇒ dynamics: independent damped oscillators forced by thermalnoise (PSD sum of Lorentzian curves). Equilibrium is canonicaland independent of damping.⇒ normal modes of reduced mass µi , resonating at ωi :

H(x, v) =1

2

∑

i

µi (ω2i x

2i + v2

i ) ; P(x, v) = e−H(x,v)/kB T/Z

Experiment: one end fixed and nearly all mass at other end. Hencenumerical simulations with µ1 ≈ M. At equilibrium, averaging overP:

〈x21 〉 =

kB T

Mω21

i.e. x1 yields a measurement of temperature.

On previous grounds, could one just use T in place of T , ingeneral, if moderately out of equilibrium?





Experiment says NO

〈x2〉 =k

BTeff

mω2

For growing gradients T separates from Teff given by spectrum!





Simulations

T T

10−2

10−1

100

101

0.6 0.8 1 1.2 1.4

ωrS

x(ω

)/

r2

0

ω/ωr





Effects of growing gradients: ∇T ↑ at same T

T T+

10−2

10−1

100

101

0.6 0.8 1 1.2 1.4

ωrS

x(ω

)/

r2

0

ω/ωr






T− T+

10−2

10−1

100

101

0.6 0.8 1 1.2 1.4

ωrS

x(ω

)/

r2

0

ω/ωr






T− T+

10−2

10−1

100

101

0.6 0.8 1 1.2 1.4

ωrS

x(ω

)/

r2

0

ω/ωr

To be publishedL. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators




Mode-mode correlations

In 1D models, a current J 6= 0 means 〈xivj〉 6= 0 for some i , j

Hyp.: For steady state, in canonical ensemble (under harmonic

approximation), Jou et al. βH ⇒ βH + γJ with

e−βH(x,v) ⇒ e−βH(x,v)−γJ(x,v); with J = − 1

N

1,N∑

i 6=k

jik xivk

γ = Lagrange multiplier of heat flux. J ∝ ∇T for small ∇T .

Guess β and make even simpler, more general, assumption on xivk :if w is one velocity correlated with x1, consider:

PNEQ(x1,w) = exp (−Mω21 x2

1/2kBT − µ w2/2kBT +λMω21 x1w)/κ

where

κ =2π√

Mω21

[µ/(kBT )2 − λ2Mω2

1

] ; and T = (T1 + T2)/2





〈x1w〉 =λ

µ/(kBT )2 − λ2Mω21

; 〈x21 〉 =

µ〈x1w〉λMω2

1kBT

Introduce

φ = −Mω21〈x1w〉 ; η =

µ

Mω21(kBT )2

; λ(φ) =1−

√1 + 4ηφ2

2φ

then⟨x2

1

⟩=

η

η − λ(φ)2

⟨x2

1

⟩(eq)(T )

with limit cases

⟨x2

1

⟩'(1 + ηφ2

) ⟨x2

1

⟩(eq)(T ) , |φ| � 1/

√η

⟨x2

1

⟩' √η|φ|

⟨x2

1

⟩(eq)(T ) , |φ| � 1/

√η

i.e.

⟨x2

1

⟩⟨x2

1

⟩eq

− 1 ∝ (∆T )2 , ∆T � T





Simulations and experiment





In experiment, normal-mode analysis justified by high Q;Fourier law by small gradients;

experimental data agree with numerical results for such simplemodel, for thermo-mechanical properties and as well as forvibrational energy of solids, as functions of T , at small ∇T ;

temperature immediately ceases to be the sole parametercharacterizing fluctuations of long-wavelength modes:indeed strong dependence of “Teff ”, i.e. of

⟨x2⟩, on ∇T ;

Experiment constitutes protocol to measure value ofLagrange multiplier λ, the “heatability” of the mode;

dependence on initial conditions?

theory and range of applicability?


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